Abstract
We develop an improved Alcock–Paczynski (AP) test method that uses the redshift-space two-point correlation function (2pCF) of galaxies. Cosmological constraints can be obtained by examining the redshift dependence of the normalized 2pCF, which should not change apart from the expected small nonlinear evolution. An incorrect choice of cosmology used to convert redshift to comoving distance will manifest itself as redshift-dependent 2pCF.
Our method decomposes the redshift difference of the two-dimensional correlation function into the Legendre polynomials whose amplitudes are modeled by radialfitting functions. Our likelihood analysis with this 2D fitting scheme tightens the constraints on Ωm and w by ∼40% compared to the method of Li et al. that uses one- dimensional angular dependence only. We alsofind that the correction for the nonlinear evolution in the 2pCF has a non-negligible cosmology dependence, which has been neglected in previous similar studies by Li et al. With an accurate accounting for the nonlinear systematics and use of full two-dimensional shape information of the 2pCF down to scales as small as 5h−1Mpc it is expected that the AP test with redshift-space galaxy clustering anisotropy can be a powerful method to constraining the expansion history of the universe.
Key words: cosmological parameters– cosmology: theory – dark energy – large-scale structure of universe
1. Introduction
The accelerated expansion of the universe remains a deep mystery that remains unsolved by contemporary cosmology.
The most popular cosmological model,ΛCDM, exquisitely fits the peaks of the cosmic microwave background(Hinshaw et al.
2013; Planck Collaboration et al. 2018), the distance–redshift relation of distant SNe1a (Perlmutter et al. 1999), and the distribution of galaxies(Li et al.2016). All the more reassuring is the small set of parameters required and the simplicity in the underlying assumptions of homogeneity, Gaussianity, and near scale invariance of the initial perturbation spectrum.
However, we are still left with the unsavory prospect that if we are to believe ΛCDM then we are forced to include within our cosmic inventory a vacuum energy that is 120 orders of magnitude smaller than theoretical predictions (e.g., Weinberg 1989) and a large component of matter that is not contained within the standard model of particle physics (e.g., Trimble 1987).
This has led many theorists to consider alternative models that include a scalar field remnant from the big bang (see Li et al.2011for review) and modifications to Einstein’s general relativity(see Koyama2016for review). Thus, the endeavor of cosmology today does not lack a rich variety of theoretical models; rather, we lack precise observations of the expansion of the universe, which would allow us discern among the proposed models.
A redshift survey is one of the most reliable ways to obtain data for uncovering the underlying cosmology of the universe.
Many statistical tools have been developed to extract informa- tion on the initial conditions of the cosmic structures from the
observed spatial distribution of galaxies. The size and depth of surveys have been improved thanks to technological advances.
Upcoming and ongoing surveys such as DESI (Dark Energy Spectroscopic Instrument; Flaugher & Bebek 2014), eBOSS (Zhao et al.2016), and PFS (Subaru Prime Focus Spectrosc- opy; Takada et al.2014) will measure the locations of millions of galaxies up to a redshift of 2 or greater, together with many of those at redshifts below 1.
In the linear regime, clusters of galaxies on very large scales retain information about the early universe, and therefore can be directly compared with the predictions of theoretical models.
If there are statistical measures of galaxy clustering that suffer from little nonlinear evolution effects either because the scale under study is safely in the linear regime or because they are intrinsically insensitive to nonlinear effects, they will be very useful for uncovering the physics of the early universe. In addition, they can be used for reconstructing the expansion history of the universe. This is because the conversion of observed redshift to comoving distance can create artificial systematic distortion of galaxy clustering when the cosmology adopted for the conversion is incorrect. Park & Kim (2010) adopted this idea and proposed using the shapes of the 2pCF, power spectrum, or the genus topology of large-scale structures in the universe as cosmic invariants for constraining the cosmological parameters governing the expansion history of the universe(see also Appleby et al.2017,2018b).
The Alcock–Paczinsky (AP) test is one of the statistical means of extracting cosmological parameters from galaxy redshift survey data(Alcock & Paczyński1979; see Figure1).
It is a test of the geometry of cosmic objects or galaxy
distribution, which should appear isotropic at all redshifts if the objects and galaxy clusters are spherical or isotropic, and the redshift of the galaxy is converted to distance from correct cosmology. For the test, the baryonic acoustic oscillation (BAO) feature in the galaxy 2pCF is often used due to its distinct excess at r∼100h−1Mpc. The AP test with BAO has been proven to put a moderately powerful constraint on parameters like the matter density parameter Ωm, dark energy equation of state w, and so on. A weakness of the BAO method is that it uses clustering information on very large scales where the statistics of a given sample is relatively weak (Li et al.
2016).
An extension of the AP test came from the observation that even though the observed galaxy clustering in the redshift space is quite anisotropic due to the redshift-space distortion effects, its redshift evolution is almost conserved, with only small nonlinear effects(Li et al.2015). The redshift evolution of the shape of the 2pCF turned out to be a powerful tool for uncovering the expansion history of the universe, as the signal of the 2pCF is much stronger near the 10 h−1Mpc scale than at the BAO scale of∼100 h−1Mpc due to a much larger number of galaxy pairs. It has been shown that the shape of the redshift- space 2pCF down to the 6 h−1Mpc scale does not evolve as much as the incorrect cosmology assumption would distort it, making it possible to separate the systematic effects due to adopting an incorrect cosmology (Li et al.2015,2016,2018;
Appleby et al.2017,2018a,2018b).
There is another stream of efforts by Ramanah et al.(2019) that tackles this problem using a Bayesian inference frame- work. Their work also aims to improve the cosmological constraint by utilizing two-point statistics of galaxies on the entire range of scales.
This work is a continuation of our effort to improve the AP test with galaxy clustering anisotropy. Previous works by Li et al.(2016,2017,2018) left room to improve the method, and we attempt to accommodate those past studies here. The main contribution of this paper is as follows.
(1) Li et al. (2016,2017,2018) used the angular dependence of the radially integrated 2pCF, which potentially dilutes the constraints from the radial shape of the 2pCF. We shall attempt
to use both angular and radial shapes of the 2pCF to place the constraints and see how much the constraints improve.
(2) A potential caveat in the previous work is that the systematic correction due to intrinsic evolution of the redshift- space 2pCF was assumed to be cosmology-independent without proof. Now that we have a data set for multiple cosmologies(see Section2.2for detail), we can verify whether or not the assumption is correct.
The rest of the paper is as follows. In Section2, we list and explain the N-body simulations and mock galaxy catalogs used for our analysis. We propose our new AP test method in detail in Section 3. We present the results in Section 4. Finally, we summarize and conclude in Section5.
2. Data 2.1. Horizon Run 4
The Horizon Run 4(HR4) simulation (Kim et al.2015) is a massive cosmological simulation that evolved Np=63003 particles in a cubic box of a side length of Lbox=
3150 h−1Mpc. It uses a flat ΛCDM cosmological model in concordance with a Wilkinson Microwave Anisotropy Probe (WMAP) 5 yr observation (Dunkley et al. 2009), where the matter density fraction, dark energy density fraction, and dark energy equation of state at z=0 are W W( m, de,w)=
0.26, 0.74, -1
( ). The volume of the HR4 is big enough to simulate the formation of large-scale structures, and at the same time its force and mass resolutions are high enough to simulate the formation of individual galaxies down to a relatively small mass scale. Thanks to these unique features, the HR4 has been extensively used for cosmological model tests and study of galaxy formation under the influence of large-scale structures in the universe (Kim et al. 2015; Hwang et al. 2016; Li et al.
2016, 2017; Appleby et al. 2017,2018b; Einasto et al. 2018;
Uhlemann et al.2018b,2018a).
Rich information on structure formation is contained in the merger trees of dark matter (DM) halos forming in the big simulation box of HR4, constructed at 75 time steps between z=12 and 0 with the time interval of ∼0.1 Gyr. In each snapshot, DM halos are found with the Friends-of-friends
Figure 1.(Left) Distribution of mock galaxies in the Horizon Run 4 simulation at z=1. (Right) Same mock galaxies with their distribution distorted by assuming Ωm=0.46 and w=−0.5, while the true cosmology has Ωm=0.26 and w=−1.
galaxies is estimated by adopting the modified merger time- scale model of Jiang et al.(2008):
= +
+
a
t
t M M
M M
0.94 0.60 0.86
ln 1 , 1
merge dyn
0.60
host sat
host sat
( )
[ ( )] ⎛ ( )
⎝⎜ ⎞
⎠⎟
where ,Mhost,Msat,tdyn are the circularity of the satellite’s orbit, mass of central and satellite halos, and the orbital period of virialized objects, respectively. We set α=1.5, which makes the 2pCF of our mock galaxies match that of the SDSS Main galaxies down to scales below 1 h−1Mpc (Zehavi et al.
2011).
For our analysis, we divide the HR4 simulation box into 6 pieces in each dimension, thereby creating 216 sub-cube mock samples that are 525 h−1Mpc long on a side. This choice is made to have a sufficient number of samples for likelihood analysis. Galaxy surveys like the SDSS cover a larger volume at the redshift of our interest(z ∼ 1). Thus, we plan to analyze larger sample volumes with larger simulations in future studies.
We adopt 10−3galaxy per(h−1Mpc)3for the galaxy number density in the mock sample, which corresponds to 0.145 million in each sub-cube mock. This number density roughly correspond to the r-band magnituder -5 logh < -20.3at z=0 (Choi et al.2010) and it is also similar to the expected number density galaxies to be observed by the PFS survey. We will also show some results with 10 times more galaxies for comparison. We note that these mass cuts are rather arbitrary.
The actual value to be used in the analysis of a given observational data should be determined by the survey data.
2.2. Multiverse Simulations
The multiverse simulations are a set of cosmological N-body simulations designed to see illustrate the effects of cosmolo- gical parameters on the clustering and evolution of cosmic structures. We changed the cosmological parameters around those of the standard concordance model with Ωm=0.26, Ωde=0.74, and w=−1. We used exactly the same set of random numbers to generate the initial densityfluctuations of all the simulations, which allow us to make a proper comparison between the models with the effects of the cosmic variance compensated.
Five multiverse simulations we use in this paper are listed in Table1. Two models have the matter density parameter shifted by 0.05 from thefiducial model, while the dark energy equation of state isfixed to w=−1. The other two quintessence models (Sefusatti & Vernizzi 2011) have w shifted by 0.5 from the fiducial value of −1, while Ωm is fixed to 0.26. These
way that the rms of the matterfluctuation linearly evolved to z=0 has σ8=0.794 when smoothed with a spherical top hat with R=8h−1Mpc.
The number of particles evolved is Np=20483 and the comoving size of the simulation box is 1024 h−1Mpc. The starting redshift is zinit=99 and the number of global time steps is 1980 with equal step size in the expansion parameter, a.
We have used the CAMB package to calculate the power spectrum at zinit. We have extended the original GOTPM code (Dubinski et al. 2004) to gravitationally evolve particles according to the modified Poisson equation of
f p r d
= + W
Ga D W
D a
4 m m 1 de a , 2
m de
m
2 2¯ ( )
( ) ( )
⎛
⎝⎜ ⎞
⎠⎟
where Dde and Dm are the linear growth factors of the dark energy and matter, respectively(see Sefusatti & Vernizzi2011 for details).
3. Methodology
3.1. Shape of the Two-point Correlation Function In our AP test we use the two-dimensional shape of the galaxy 2pCF in the plane of the line of sight and across the line- of-sight directions. However, we exclude the region r^<
^ º -
r ,cut 3h 1Mpc of the plane from our analysis to minimize the impact of the highly nonlinear Fingers-of-God effects. This leaves us with only a mildly nonlinear contribution to x in our analysis.
Because we use only the shape of x and not its amplitude in the AP test, we shall normalize x by its volume integral up to the radial separation of rmax=45 h−1Mpc. Namely, the normalization factor is
ò
mò
x z º 2p d r drx r,m,z , 3
r 0
1 0
max 2
*( ) ( ) ( ) ( )
where r is radial separation of galaxy pair andμ is the cosine of the angle between the line of sight and pair-separation direction. Then, the normalized 2pCF is
x m x m
º x
r z r z
, , , z,
. 4
*
ˆ ( ) ( )
( ) ( )
Regarding the choice of rmax=45 h−1Mpc, we picked a value that is large enough to make the normalization insensitive to the highly nonlinear small-scale physics of the Fingers-of-God effect. We find our results are generally insensitive to the
choice of rmax. The left panel of Figure2shows xˆ from one of the sub-cube mock galaxy samples.
In this work, we use the Legendre polynomials P0=1, m
= -
P2 (3 2 1) 2 and P4=(35m4-30m2+3) 8 to approximate the angular dependence of xˆ at each r:
x m x m
x m x m
=
+ +
r z r z P
r z P r z P
, , ,
, , . 5
0 0
2 2 4 4
ˆ ( ) ˆ ( ) ( )
ˆ ( ) ( ) ˆ ( ) ( ) ( )
Here, xˆ , x0 ˆ , and x2 ˆ are similar to the monopole, quadrupole,4 and hexadecapole moments at a given r, except we exclude
^< -
r 3h 1Mpc in the fitting. In this case, we cannot use a decomposition formula like xˆℓ=
ò
x mˆ ( ) ( )Pℓ m md and have to make aχ2fitting instead. We write xˆ0,2,4 when we refer to xˆ ,0 xˆ , and x2 ˆ altogether in the rest of the paper. Thanks to this4 exclusion of the highly nonlinear part of the 2pCF, thefits by these three moments are highly accurate (Figure 2). In principle, xˆ can be decomposed into Legendre polynomials of arbitrary order, but wefind that higher-order moments do not help much with tightening the constraint.The cosmic variance infinite survey volume would give an intrinsic scatter to xˆ0,2,4. The uncertainty ranges of those moments are computed from the 216 sub-cube mock samples of HR4 and are shown as shaded regions in Figure 3. For our main analysis, we shall use the differences xˆ across different redshifts, which measure the shape change of 2pCF between redshift. We shall introduce the difference,Dˆ, in Sectionx 3.4, along with our motivation for adopting it.
3.2. Geometrical Distortion Effects due to Choice of Incorrect Cosmology
If an incorrect cosmology is adopted when converting redshift to distance, then the apparent spatial distribution of galaxies will be distorted. Recalling that the comoving displacements are D = Dr z c H z[ ( )]andD =r^ (1 +z D z) A( )Dq, for an object subtending Δz and Δθin the parallel and perpendicular to the line-of-sight direction, respectively, the distortion in each
direction can be parameterized by a
a
=
^ =
-
z H z
H z
z D z
D z . 6
adopted true A,adopted
A,true 1
( ) ( )
( )
( ) ( )
( ) ( )
⎡
⎣⎢ ⎤
⎦⎥
In the case of ourfiducial cosmology (Ωm, w)=(0.26, −1), if we adopt “incorrectly” that (Ωm, w)=(0.41, −0.5) then a = 0.735 and a =^ 0.841 at z=1. Relative to the reference point this is a −26.5% and −15.9% change in r and r ,^ respectively(see Figure1), which makes the apparent shape of the galaxy distribution compressed relatively more along the line of sight.
The distortion effect in the 2pCF due to adopting incorrect cosmology is described by Li et al. (2016). The effect is described by the coordinate transformation(Li et al.2018)
x¢ ¢(r^,r¢ =) x(r^ a^,r a). ( )7 In polar coordinates, the transformation is x¢ ¢(r,m¢ =) x(r,m), where
a m a m
m m
a a m a m
= ¢ ¢ + - ¢
= ¢ ¢ + - ¢
- ^-
- ^-
r r 1
1 1
. 8
2 2 2 2
2 2 2 2
( )
( )
( )
We normalize the 2pCF after the transformation, as in Equations(3) and (4).
Using the transformation, we show how xˆ0,2,4 is affected by choosing incorrect cosmology. In the left panel of Figure3, the black solid lines are xˆ0,2,4 for the fiducial case of (Ωm, w)=
(0.26, −1) and the other lines are those with the distortion effect applied for four incorrect cosmologies(Ωm, w)=(0.26,
−0.5), (0.26, −1.5), (0.21, −1), and (0.31, −1). The significance of the distortion effect appears strongest for xˆ at2
separations roughly between 7 and 15 h−1Mpc. In the incorrect cosmologies, xˆ fall nearly outside the 22 σ uncertainty in that range. The distortion effect is smaller for xˆ and x0 ˆ , but it does4 seem to be significant for certain separations.
Figure 2.Left: normalized correlation function xˆ of mock galaxies at z=1. On the left of the vertical dashed line atr^=3h-1Mpcis the part that we exclude from our analysis. The arcs denote circles of r=6 (red) and 30h−1Mpc(black), on which we show the angular shape of xˆ in the right panel. Right: ξ(r, μ) is shown as a function of the cosine of the angle between the line of sight and the separation direction,μ, for two pair separations, r=6 (red) and 30h−1Mpc(black). The dots and thin solid lines are the data points and the thick dashed lines are ourfit for the data.
3.3. Cosmology Dependence in the Shape of a Two-point Correlation Function
If the amount of redshift distortion of xˆ0,2,4 remained the same for different cosmologies, we would be able to use xˆ0,2,4 to constrain the cosmology using the corrections for nonlinear evolution effects found for just one cosmology. However, xˆ0,2,4 does have some cosmology dependence, as we shall show in this section.
In the right panel of Figure3, we show xˆ0,2,4for(Ωm, w)=
(0.26, −1), (0.26, −0.5), (0.26, −1.5), (0.21, −1), and (0.31,
−1) at z=1, which are calculated from the multiverse simulation set. In comparison to the left panel, where we plotted the effect of an incorrect cosmology choice, we describe the intrinsic cosmology dependence of xˆ0,2,4in the right panel.
The cosmology dependence is tiny for xˆ , but it is signi4 ficantly large for xˆ and x0 ˆ reaches nearly the 22 σ level for certain cases.
In the next section it is shown that the redshift evolution of xˆ0,2,4also depends on cosmology, and this is what needs to be corrected.
3.4. Evolution of the Shape of the Two-point Correlation Function
Our AP method does not care whether or not the amplitude or shape of the correlation function depends on cosmology as it cares only about whether or not the function changes across redshifts. We shall show that the redshift evolution of 2pCF, namely Dxˆ0,2,4(z zi, j)ºxˆ0,2,4( )zi -xˆ0,2,4( )zj between two
different redshifts for example, is not sensitive to the under- lying cosmology in this section.
We computeDxˆ0,2,4(z zi, j) for zi=1 and zj=0 for every possible pair of mocks of the total 216 HR4 mocks, excluding the cases that use the same mock for both xˆ0,2,4(zi=1) and xˆ0,2,4(zj=0). Using the same volume for z=1 and 0 would underestimate the scatter in Dˆ because the cosmic variancex will be mostly canceled out. We thus have 2153 cases of
x
Dˆ0,2,4(z zi, j), which we plot in Figure 4 with the effect of incorrect cosmology choice in the left panel and intrinsic cosmology dependence in the right panel.
Dˆ is very small compared to xˆ (see Figurex 3). This was also addressed in Li et al.(2016). Therefore, even though the shape of the 2pCF itself is significantly distorted due to the redshift- space distortion effect, it is roughly a cosmic invariant and can be used for the AP test. In reality, the shape of the redshift- space 2pCF mildly evolves due to nonlinear gravitational evolution and the change of the type of galaxies. Due to the nonlinear effect,Dxˆ shows a nonzero residual that decreases2 with r. However, we can see from the left panel of Figure4that this change is smaller compared to that produced by the geometrical distortion caused by adopting an incorrect cosmology(see also Li et al.2016).
The cosmology dependence of Dˆ appears significantlyx smaller than that of xˆ. There is more than a 2σ level of change in xˆ when changing Ω0 mfrom 0.26 to 0.21 or 0.31(Figure3), but that in Dxˆ0 is well below the 1σ uncertainty at most
Figure 3.Left:r2xˆ (0 r z, =1)(top), xr2ˆ (2r z, =1) (middle), and xr2ˆ (4r z, =1)(bottom) for the fiducial model of HR4 are plotted as black solid lines. The uncertainty ranges of those values for(525 h−1Mpc)3volume are shown in dark gray(1σ range) and gray (2σ range) based on the results from the 216 sub-cube mock samples of HR4 with 10−3galaxies per(Mpc/h)3. Each sub-cube contains 0.145 million galaxies. The red, blue, and black dashed, and black dotted lines are the results of distorting the spatial galaxy distribution in thefiducial case by incorrectly assuming (Ωm, w)=(0.26, −0.5), (0.26, −1.5), (0.21, −1), and (0.31, −1), respectively. Right: the lines are the results of each of the multiverse simulations. The lines now describe the intrinsic shape of moments in each cosmology.
separations. The change in xˆ ranges from the 12 σ to 2σ level, but that inDxˆ is mostly below 12 σ. Both xˆ and x4 Dˆ do not4 seem to be affected by background cosmology to a significant level.
However, the residual cosmology dependence inDˆ is notx negligibly small everywhere and it has to be taken in account.
We expect the 1σ level change in Dxˆ2 at separations 5∼10 h−1Mpc to significantly affect the results (see the middle right panel of Figure 4). That is where we expect to have the strongest constraint from the geometrical distortion effect. We describe how we subtract this effect in Section 3.6.
3.5. Parameterization of the Redshift Evolution of 2pCF In order to use Dˆ for the AP test, we need to compressx the information inDˆ into a small number of parameters. If thex number of parameters is comparable to that of the samples, the covariance matrix will be significantly biased and the error will propagate to the likelihood evaluation (Hartlap et al. 2007;
Percival et al.2014).
We fit the r-dependence of the moments, Dxˆ0,2,4, with second-order polynomials as follows:
x x x
D = + +
D = + +
D = + +
- - -
r z z r a a r a r
r z z r a a r a r
r z z r a a r a r
, , log log
, , log log
, , log log . 9
f i j
f i j
f i j
0, 2
1 2 3 2
2, 2
4 5 6 2
4, 2
7 8 9 2
ˆ ( ) ( [ ( )] [ ( )] )
ˆ ( ) ( [ ( )] [ ( )] )
ˆ ( ) ( [ ( )] [ ( )] ) ( )
Thisfitting results in nine parameters ºa (a a1, 2, ¼, a9) that describeDˆ. Namelyx
å
x x m
D » D
=
r P . 10
ℓ
ℓ f ℓ
0,2,4
ˆ ˆ , ( ) ( ) ( )
We shall use this nine-element vector a to describe Dˆ tox calculate the likelihood for each cosmology. Wefind that the constraint is strongest when we fit between r=5 and 15 h−1Mpc. At r> 15 h−1Mpc xˆ does not contribute to the constraint of the cosmology and we exclude it from our analysis. We thus use that range to generate a.
3.6. Likelihood Analysis
The redshift difference in the shape of 2pCF,Dˆ, is muchx smaller than the shape itself (xˆ), but it does have a nonzero residual, as can be seen from the right panel of Figure4. We shall refer to the value as the systematics and use the superscript“sys” to denote it. Our goal is to make an accurate subtraction of asysto the observed value of a in our likelihood analysis.
x
Dˆ0,2,4(z zi, j) is cosmology-dependent, as shown in the right panel of Figure 4. Li et al. (2016,2017, 2018) assumed it is cosmology-independent, which is a potential caveat of their studies.
In principle, asys should be computed for every cosmology under consideration, which would be too expensive. In this work we linearly interpolate and extrapolatefive cases of asys
Figure 4. Same as Figure 3 except we plot the redshift evolution between z=0 and 1, r2Dx r z, i=0,zj=1
ˆ (0 ), r2Dx r z, i=0,zj=1
ˆ (2 ), and
x
D = =
r2 r z, i 0,zj 1
ˆ (4 ) in the upper, middle, and lower panels, respectively. The black solid lines show the nonlinear evolution of each multipole between z=0 and 1 in the case of the fiducial cosmology, (Ωm, w)=(0.26, −1). On the right panels it is shown that the amount of redshift evolution depends weakly on cosmology.
parameter space to be fairly low and have minimal impact on high-likelihood region. We use the systematics-correctedfitting parameters
W º W - W
p( m,w) a( m,w) asys( m,w) (11) for the likelihood calculation.
Then, we calculate the covariance matrix, ij, using p from 2162combinations of HR4 sub-cube mocks at z=0 and 1 for the fiducial cosmology (Ωm, w)=(0.26, −1). This matrix describes the uncertainty range of Dˆ in the fiducialx cosmology.
Next, we compute p for arbitrary cosmology. For an adopted cosmology, we transform xsysusing Equation(8) and compute a from it. This would put the center of our constraint at the fiducial cosmology Ωm=0.26 and w=−1. Then,
= -
p a asys contains the distortion effect from incorrect cosmology choice, but not the cosmic variance. Finally, the likelihood for any adopted cosmology is =exp(-c2 2), where
åå
c Wm,w º p W ,w p W ,w. 12
i j
i m ij j m
2( ) ( ) · · ( ) ( )
We shall show the likelihood results from the above equation in the next section.
4. Results: Cosmological Constraints 4.1. Constraints from Different Redshift Intervals The likelihood contour forΩm−w from our AP test is shown in Figure 5. The size of the contour is our prediction for the constraining power of our method. As mentioned above, the center of the constraint is designed to be at the fiducial cosmology. In an actual analysis of observational data from the fiducial cosmology, the center of constraint will be located randomly within the range of uncertainty (i.e., the area enclosed by the contour).
The constraint from the evolution between z=1 and 0 ( xDˆ (zi=1,zj=0 ;) top left panel) forms a stretched region.
The direction of the stretch is similar to the line that satisfies a=a^ in the parameter space. This is because the AP test loses the constraining power when the ratio of distortions parallel and perpendicular to the line-of-sight direction, a a^, remains unchanged.
The likelihood contour forDxˆ (zi=0.5,zj=0)(upper right panel of Figure 5) has a similar stretched shape, but is more tilted toward the horizontal direction. This is because the distortion factors aand a^have a different dependence onΩm
of ΔΩm≈0.04 or Δw≈0.2 when marginalized over w or Ωm, respectively.
4.2. Impact of Cosmology Dependence in the Systematics In previous works by Li et al. (2016, 2017, 2018), the systematics was modeled from a single background cosmology and the same correction is made for the entire range of cosmology considered. Here, we assess the impact of accounting for the cosmology dependence of the systematics correction, asys. In Figure 5 we show the cosmological constraints when afixed systematics correction of afid
sys is used regardless of adopted cosmology (yellow and magenta contours).
The likelihood contour is much more stretched for fixed systematics correction cases, showing stronger degeneracy along the line of a^=a. For the combined constraint, the shape is more elongated forfixed systematics, but the area of the contour is not very affected: ignoring the cosmology dependence of the systematics correction underestimated the uncertainty in the parameter estimation only slightly (20%).
However, we note that if the cosmology dependence is not taken into account, the central value of the constraint is likely to shift in the analysis of real observational data by more than 1σ, as shown in xDˆ across different cosmologies in the right panel of Figure 4. Our likelihood results are designed to be centered at thefiducial cosmology.
The cosmology dependence in the systematics may seem to help break the degeneracy for parameter sets that are far from ourfiducial choice (Ωm=0.26 and w=−1). However, the systematics correction outside the coverage of the multi- verse simulations(see the diamond in Figure 5) involves less reliable extrapolation and we cannot draw conclusions for those parameters in this work. More simulations are needed to extend the coverage. In practice, the result is unlikely to change significantly due to the extra simulations, as other cosmological probes like the CMB show that the constrained area will be within the range of multiverse simulations.
4.3. Dependence of Number Density and Type of Galaxies The constraining power of our AP test is determined by the size of the uncertainty in Dxˆ0,2,4, which is quantified by the covariance matrix in Equation (12). The uncertainty range is also described as the shaded areas in Figure4. The smaller the shades are, the stronger the constraint is. This uncertainty is for ourfiducial sample that is (525 Mpc/h)3in volume with 10−3 galaxies per(Mpc/h)3.
The uncertainty is presumably from cosmic variance infinite volume and Poisson noise due to a limited number of pairs. In that case, one can reduce the uncertainty by increasing either the survey volume or the number density of sample galaxies. In this section, we show the impact of increasing the sample density by 10 times. The denser sample contains about 1.45 million galaxies and includes relatively less massive galaxies.
We note that having such a dense sample from near future surveys is not practical for redshifts of our interest (z0.5), while we will likely have a larger volume than our mock from those surveys. We shall explore the impact of enlarged survey volume in future works with larger simulations.
In Figure6, we compare the uncertainty range of ourfiducial casen=10-3(h-1Mpc)-3with another sample with 10 times higher galaxy number density n=10-2(h-1Mpc) . In the-3 case with higher number density, the uncertainty is substan- tially reduced forDxˆ and x2 Dˆ at4 10h-1Mpc, where most of the constraint comes from. As a result, the predicted cosmological constraint in the high number density case turns out be substantially tighter. It gives a nearlyfive times smaller 1σ (2σ) area and more than two times tighter constraint for each parameter(see Figure7), giving marginalized constraints ofΔΩm≈0.017 or Δw≈0.09 with 1σ uncertainty. Note that this is more than a factor of two improvement compared to the result of thefiducial case, ΔΩm≈0.04 and Δw≈0.2.
Note that the average distance between galaxies is 10 (h−1Mpc) for n=10-3(h-1Mpc) , while it is-3 ∼4.6 (h−1Mpc) for n =10-2(h-1Mpc) . The uncertainty is-3 significantly reduced at 10h−1Mpc as we increase the number density, while it stays nearly the same at larger scales.
The uncertainty seems to be dominated by the cosmic variance on scales larger than the mean galaxy separation, while the shot noise seems to dominate on scales shorter than the mean separation. Also, it can be seen in Figure 6 that the size of systematics is larger for the less massive galaxies, particularly forDxˆ . Therefore, it is necessary to estimate the systematics2 and have the abundance of galaxies matched with observations.
4.4. Comparison with the Previous Method
Li et al. (2016,2017,2018) used the binned values of the radially integrated 2pCF, which is similar to
ò ò
x m
x m x m
D
º -
D z z
r z dr r z dr
, ,
, , , , , 14
r i j
r r
i r
r
j
min max
min max
ˆ ( )
ˆ ( ) ˆ ( ) ( )
wherermin=5h-1Mpc andrmax=45h-1Mpc are chosen in their AP test. Their method potentially suffers from loss of information in the radial shape of the 2pCF. Note that the radial
Figure 5.Likelihood map(Wm,w) for the cosmological constraint fromDˆ (x zi=1,zj=0)(upper left), xDˆ (zi=0.5,zj=0)(upper right), and the combined constraint from both(lower). The black and gray contours enclose the 1σ and 2σ ranges of the constraint, respectively. In the case in the lower panel, we plot the marginalized constraint for each parameter asfilled curves attached to axes with the same color convention for the contours. The yellow solid and magenta dashed lines enclose the 1σ and 2σ regions, respectively, assuming a fixed systematic correction ofasys=afidregardless of assumed cosmology. The cyan dotted lines in the upper panels describe the parameter sets that give a^=a. The diamond symbols denote thefive sets of Ωmand w covered by the multiverse simulations.
integrals on the right side of Equation (14) are practically dominated byξ at r=rminbecause of the r−2-like scaling ofξ.
We expect the constraining power of the AP test to improve with our method, which uses both the radial and angular dependence ofξ. To compare the constraining power of the two methods, we reproduce their AP test with nine μ-bins of
x m
DˆDr( ,zi=1,zj=0) and with rmin=5h−1Mpc and rmax=45h−1Mpc. We set the number of μ-bins to be the same as the number of parameters we use to fit xDˆ0,2,4. The systematics correction is assumed to be cosmology-indepen- dent in those works. That is, asys =afidsysfor all cosmologies.
The resulting likelihood for Ωm−w is shown in Figure 8.
The constraint from the method of Li et al.(2016,2017,2018) gives about a 40% larger uncertainty in the parameter estimation(black and gray areas). Clearly, using the full shape improves the constraint by a significant amount.
The amount of improvement in constraint, however, may not look very impressive considering that we added an extra dimension in the analysis. This is because the geometric distortion effect inDxˆ0,2,4over different r is correlated to some extent. As we see in the left panel of Figure 4, incorrect cosmology choice shifts or tiltsDxˆ0,2,4more or less uniformly
over r. In this case, combining constraints over different r does not add up perfectly. Nevertheless, using the full shape does improve the constraint by a significant amount.
5. Summary and Conclusions
The AP test that uses the evolution of redshift-space 2pCF as proposed by Li et al. (2015, 2016) is a powerful method for constraining the cosmological parameters governing the expansion of the universe. We presented an improved method for the AP test that utilizes the two-dimensional shape of the anisotropic galaxy clustering down to a scale as small as
Figure 6. Uncertainty range of Dxˆ (0zi=1,zj=0) (upper panel), x
Dˆ (2 zi=1,zj=0) (middle panel), and Dxˆ (4zi=1,zj=0) (lower panel) with two different sample galaxy number densities, n=10−2 and
- h- -
10 3( 1Mpc) . The dark and light gray shade describes and 13 σ and 2σ uncertainty ranges, respectively, for a mock sample with n=
- h- -
10 2( 1Mpc)3 and the pairs of blue solid and black dashed lines in each panel describe 1σ and 2σ uncertainty ranges, respectively, for n=
- h- -
10 3( 1Mpc) .3
Figure 7. Constraint from 10 times more galaxies in the same volume.(Wm,w) from a combined constraint ofDxˆ (zi=1,zj=0) and
x
Dˆ (zi=0.5,zj=0), with a galaxy number density ofn=10-2(h-1Mpc)-3 is shown with black (1σ) and gray (2σ) filled contours. The marginalized constraint for each parameter is shown asfilled curves on each axis. The yellow solid and magenta dashed line contours show the constraint from ourfiducial galaxy number density ofn=10-3(h-1Mpc) , which is the same as the-3 black and grayfilled contours of Figure5.
Figure 8. Constraint from the method of Li et al. (2016). (Wm,w), a combined constraint ofDxˆDr(zi=1,zj=0) andDxˆDr(zi=0.5,zj=0) from Equation (14), is shown in black (1σ) and gray (2σ) filled contours. For comparison, the constraint from this work assuming cosmology-independent systematics is shown with yellow solid(1σ) and magenta dashed (2σ) contours.
Note that the line contours are the same as those in the lower panel of Figure5.
5 h−1Mpc. We also showed the importance of accounting for the cosmology dependence of the systematics correction, which has been neglected in previous works. In this work we focused on describing and justifying the method with ideal mock galaxy samples constructed from a high-resolution, large-volume N- body simulation. We shall apply this methodology to observational data in future works.
Our method decomposes the two-dimensional galaxy 2pCF into Legendre polynomials whose amplitudes are modeled by radial fitting functions (Equations (5) and (9)). This allows us to describe the 2D shape of the 2pCF with a reasonably small number of parameters. Our likelihood analysis with this 2D fitting scheme tightens the constraint on Ωm and w by 40% compared to the previous method of Li et al.
(2016,2017,2018) that uses one-dimensional angular depend- ence only.
We found that the systematic effects in the shape of 2pCF have a non-negligible amount of cosmology dependence over Ωm=0.21–0.31 and w=−0.5–1.5, which can result in changes in the shape of the constraint. The cosmology dependence is likely to change the center of constraint in the case of observational data. Therefore, future works should account for cosmology dependence with more simulations from different background cosmologies.
The constraint on Ωm and w from a single pair of redshifts has a degeneracy for the parameter sets that give the same a a^ . This degeneracy can be broken by adding extra pairs of redshifts in the analysis. Most of the constraint comes from the smallest scales we consider, between r=5 and 10 h−1Mpc.
Reducing the uncertainty in the shape of 2pCF at those scales is the key to tightening the constraint; this can be achieved by increasing the number of galaxy pairs or enlarging the survey volume. When we increased the mock galaxy number density from n=10−3to10-2(h-1Mpc)-3while the physical size of the sample was fixed to (525h-1Mpc) , the constraint3 tightened by nearly two times. However, enlarging the sample volume is more relevant to the expected outcome of upcoming surveys, which we shall explore in future works.
The authors thank C. Pichon, H. S. Hwang, E. Komatsu, D. Jeong, and T. Sunayama for the helpful comments on this work. X.D.L. acknowledges the support from an NSFC grant (No. 11803094). C.G.S. acknowledges financial support from the National Research Foundation(NRF; #2017R1D1A1B0303 4900, #2017R1A2B2004644 and #2017R1A4A1015178).
S.E.H. was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education(2018R1A6A1A06024977).
The authors acknowledge the Korea Institute for Advanced Study for providing computing resources (KIAS Center for
Advanced Computation Linux Cluster System). This work was supported by the Supercomputing Center/Korea Institute of Science and Technology Information, with supercomputing resources including technical support (KSC-2016-C3-0071), and the simulation data were transferred through a high-speed network provided by KREONET/GLORIAD.
ORCID iDs
Hyunbae Park(박현배) https://orcid.org/0000-0002- 7464-7857
Cristiano G. Sabiu https://orcid.org/0000-0002-5513-5303 Sungwook E. Hong(홍성욱) https://orcid.org/0000-0003- 4923-8485
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